Questions: Solve the following system of equations. 3x + 5y - z = 1 4x + 7y + z = 2 7x + 12y = 3 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. There is one solution, 1. B. There are infinitely many solutions. The solution is (, , z), where z is any real number. C. There is no solution. Which of the following is a possible solution to the system? A. (33,-19,3) B. (44,-35,7) C. (17,4,6) D. (25,-17,7) E. All of the above are possible solutions.

Solution Steps

The given system of equations is: \[ \begin{cases} 3x + 5y - z = 1 \quad (1) \\ 4x + 7y + z = 2 \quad (2) \\ 7x + 12y = 3 \quad (3) \end{cases} \]

Add equations (1) and (2) to eliminate \( z \): \[ (3x + 5y - z) + (4x + 7y + z) = 1 + 2 \] Simplify: \[ 7x + 12y = 3 \quad (4) \]

Equation (4) is identical to equation (3): \[ 7x + 12y = 3 \] This means the system is dependent, and there are infinitely many solutions.

From equation (1), solve for \( x \) in terms of \( y \) and \( z \): \[ 3x + 5y - z = 1 \implies 3x = 1 - 5y + z \implies x = \frac{1 - 5y + z}{3} \] Substitute \( x \) into equation (3): \[ 7\left(\frac{1 - 5y + z}{3}\right) + 12y = 3 \] Simplify: \[ \frac{7 - 35y + 7z}{3} + 12y = 3 \] Multiply through by 3: \[ 7 - 35y + 7z + 36y = 9 \] Combine like terms: \[ 7 + y + 7z = 9 \implies y + 7z = 2 \implies y = 2 - 7z \] Substitute \( y = 2 - 7z \) back into the expression for \( x \): \[ x = \frac{1 - 5(2 - 7z) + z}{3} = \frac{1 - 10 + 35z + z}{3} = \frac{-9 + 36z}{3} = -3 + 12z \] Thus, the solution is: \[ (x, y, z) = (-3 + 12z, 2 - 7z, z) \]

Check each option to see if it satisfies the system:

• A. \((33, -19, 3)\): \[ x = -3 + 12(3) = 33, \quad y = 2 - 7(3) = -19 \] This satisfies the system.
• B. \((44, -35, 7)\): \[ x = -3 + 12(7) = 81 \neq 44 \] This does not satisfy the system.
• C. \((17, 4, 6)\): \[ x = -3 + 12(6) = 69 \neq 17 \] This does not satisfy the system.
• D. \((25, -17, 7)\): \[ x = -3 + 12(7) = 81 \neq 25 \] This does not satisfy the system.

Only option A is a valid solution.

The system has infinitely many solutions, and the correct choice is: \[ \text{B. There are infinitely many solutions. The solution is } (-3 + 12z, 2 - 7z, z), \text{ where } z \text{ is any real number.} \]

Final Answer